Planes in cubic fourfolds

Abstract

We show that the maximal number of planes in a complex smooth cubic fourfold in ${\mathbb{P}}^5$ is $405$, realized by the Fermat cubic only; the maximal number of real planes in a real smooth cubic fourfold is $357$, realized by the so-called Clebsch--Segre cubic. Altogether, there are but three (up to projective equivalence) cubics with more than $350$ planes.

Revise version accepted for publication. Newer references and a bound for nodal cubics added

References in corpus (1)

• 800 conics in a smooth quartic surface